Divide using synthetic division $$ ( x^{5}-23x^{3}+6x^{2}+112x-96 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-5&1&0&-23&6&112&-96\\& & -5& 25& -10& 20& \color{black}{-660} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{2}&\color{blue}{-4}&\color{blue}{132}&\color{orangered}{-756} \end{array} $$The solution is:
$$ \frac{ x^{5}-23x^{3}+6x^{2}+112x-96 }{ x+5 } = \color{blue}{x^{4}-5x^{3}+2x^{2}-4x+132} \color{red}{~-~} \frac{ \color{red}{ 756 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&-23&6&112&-96\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-5&\color{orangered}{ 1 }&0&-23&6&112&-96\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&-23&6&112&-96\\& & \color{blue}{-5} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrrr}-5&1&\color{orangered}{ 0 }&-23&6&112&-96\\& & \color{orangered}{-5} & & & & \\ \hline &1&\color{orangered}{-5}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 25 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&-23&6&112&-96\\& & -5& \color{blue}{25} & & & \\ \hline &1&\color{blue}{-5}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -23 } + \color{orangered}{ 25 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrr}-5&1&0&\color{orangered}{ -23 }&6&112&-96\\& & -5& \color{orangered}{25} & & & \\ \hline &1&-5&\color{orangered}{2}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 2 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&-23&6&112&-96\\& & -5& 25& \color{blue}{-10} & & \\ \hline &1&-5&\color{blue}{2}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrr}-5&1&0&-23&\color{orangered}{ 6 }&112&-96\\& & -5& 25& \color{orangered}{-10} & & \\ \hline &1&-5&2&\color{orangered}{-4}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&-23&6&112&-96\\& & -5& 25& -10& \color{blue}{20} & \\ \hline &1&-5&2&\color{blue}{-4}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 112 } + \color{orangered}{ 20 } = \color{orangered}{ 132 } $
$$ \begin{array}{c|rrrrrr}-5&1&0&-23&6&\color{orangered}{ 112 }&-96\\& & -5& 25& -10& \color{orangered}{20} & \\ \hline &1&-5&2&-4&\color{orangered}{132}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 132 } = \color{blue}{ -660 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&-23&6&112&-96\\& & -5& 25& -10& 20& \color{blue}{-660} \\ \hline &1&-5&2&-4&\color{blue}{132}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -96 } + \color{orangered}{ \left( -660 \right) } = \color{orangered}{ -756 } $
$$ \begin{array}{c|rrrrrr}-5&1&0&-23&6&112&\color{orangered}{ -96 }\\& & -5& 25& -10& 20& \color{orangered}{-660} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{2}&\color{blue}{-4}&\color{blue}{132}&\color{orangered}{-756} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}-5x^{3}+2x^{2}-4x+132 } $ with a remainder of $ \color{red}{ -756 } $.